Torsion of a Thin Tube under the Action of Two Torques

Let us consider a shaft of length L, diameter d, loaded with two torques T of inverse directions, applied to either of the ends. The cross-section of the shaft is a circle of diameter d.

Torsion of a shaft under the action of two torques,

Sought quantity is the maximum angle of twisting.
Assume T = 100 N*m, L = 0.5 m, d1 = 0.056 m, d2 = 0.064 m; (d1+ d2 )/2=4 mm.
Characteristics of material have values: Shear modulus G = 8.203E+010 Pa, Poisson's ratio ν = 0.28.
One of the ends is free (not fixed), and other is partially restricted in cylindrical coordinate system (only rotation about the axis is allowed, but this is not sufficient to determine model in space and therefore to complete calculation). This is done to facilitate examining the results. Both ends are loaded with torque T, pointing in opposite directions to each other.
To solve this study in AutoFEM Analysis, it is necessary to turn on the option "Stabilize the unfixed model" with additional stiffness equal 1.
You should check this box at the Properties dialog of Static Analysis on the page "Solving".

Torsion of a shaft under the action of two torques, the finite element model with applied loads and restraints

The finite element model with applied loads and restraints

The analytical solution appears as:
φ = (T*L)/(G*Jp) = 8.9425E-004 rad

u = 2.6828E-005 m

where φ - angle of twist, u - point displacement, T – is the torque, L – the shaft length, G – the material Shear modulus,

– polar moment of inertia of the cross-section.

After carrying out calculation with the help of AutoFEM, the following results are obtained:

(tube deflection is equal to (1.3522E-005)+(1.3522E-005)=2.7044E-006 m)

Table 1.Parameters of the finite element mesh

Finite Element Type	Number of Nodes	Number of Finite Elements
linear triangle	2200	4336

Table 2. Result "Displacement, magnitude"*

Numerical Solution Displacement u*, m	Analytical Solution Displacement u, m	Error δ =100%* \|w* - w\| / \|w\|
2.6828E-005	2.7044E-006	0.81

Torsion of a shaft under the action of two torques, Result "Displacement, magnitude"

*The results of numerical tests depend on the finite element mesh and may differ slightly from those given in the table.